Apparatus for multi-level encoding

ABSTRACT

The present disclosure relates to an apparatus and method for multi-level encoding of an input message sequence into a symbol sequence, for instance, based on polar coding. The input message comprises information bits. The apparatus is configured to divide the input message into a plurality of sub-messages, encode each of the sub-messages into a codeword, wherein a set of the sub-messages is encoded on the basis of the codewords obtained by encoding the sub-messages not in the set and a predefined function of the symbol sequence, and map the encoded sub-messages into corresponding symbols.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of International Application No.PCT/EP2019/079266, filed on Oct. 25, 2019, the disclosure of which ishereby incorporated by reference in its entirety.

TECHNICAL FIELD

The present disclosure relates, in general, to the field of encodingmessages. In particular, the present disclosure relates to an apparatusand method for multi-level encoding of an input message into a symbolsequence, for instance, encoding based on polar coding. The disclosurerelates also to an apparatus and method for multi-level decoding of aninput symbol sequence. The disclosure is also concerned with providingthe symbol sequence with a predefined characteristic.

BACKGROUND

In order to achieve the capacity of a transmission channel, the channelinput symbols should have a certain probability distribution. Forexample, a Gaussian distribution is required to achieve the capacity ofthe AWGN channel with average transmit power constraint. However, inmany practical systems, uniformly distributed channel input symbols areused, which cause a gap to the capacity. This loss is called the shapingloss, and can be up to 1.53 dB on AWGN channels, if uniformlydistributed channel input symbols are used.

The shaping loss becomes significant especially for high-ordermodulation. In many systems, binary coded modulation is used, whereininput messages are first mapped to binary codewords (using a channelencoder), and the codewords are then further mapped to channel inputsymbols (such as Amplitude Shift Keying (ASK) or Quadrature AmplitudeModulation (QAM) symbols) via a symbol mapper. A symbol mapper takes ingeneral m bits as input and converts them into a channel input symbol.

Usually, binary codewords have a uniform distribution of bits, i.e., thenumber of ones and zeros within a codeword is approximately the same onaverage. This causes the channel input symbols to also have a uniformdistribution, which is the cause of the shaping loss, as for manychannels uniformly distributed channel input symbols are not optimal.

Different methods to reduce the shaping loss (i.e., shaping methods) areknown, such as Probabilistically Shaped Coded Modulation (PSCM) whichcan be realized based on Bit-Interleaved Coded Modulation (BICM) orMulti-Level Coding (MLC).

In BICM a message d is first encoded by a channel encoder to a code wordc, which is interleaved and then mapped to channel input symbols x via asymbol mapper. At the receiver, a single stage demapping and decodingcan be performed (see FIG. 1).

In MLC a message d is first divided into m parts (d₁, . . . , d_(m)),wherein each part is then encoded with a different channel encoderresulting in codewords (c₁, . . . , c_(m)) of the same lengths. A symbolmapper maps each bit from the m codewords into a different bit-level ofa modulated symbol. At the receiver, each bit-level is demapped anddecoded successively (i.e. multi-stage demapping), using the informationobtained from decoding the previous bit-levels (see FIG. 2 and FIG. 3for MLC transmitter and receiver, respectively).

In the work of Iscan et al. “Probabilistically Shaped Multi-Level Codingwith Polar Codes for Fading Channels,” In 2018 IEEE Globecom Workshops(GC Wkshps), pp. 1-5, a polar coded PSCM scheme based on MLC ispresented. In FIG. 4 a plot of the block diagram of this scheme isshown.

An example of the resulting probability distribution is given in FIG. 5.It can be observed that symbols near origin (which have low energy) havea higher probability than the symbols that are away from the origin(which have a high energy). This method is called ‘single bit-levelshaping’. It can be further observed that the resulting probabilitydistribution is only a coarse approximation of the (optimal) Gaussiandistribution. Therefore, this method only allows to obtain a limitedshaping gain.

The following facts can be summarized about this conventional approach:

-   -   The scheme in the work of Iscan et al. encodes each bit-level        independently, and only a single bit-level is shaped.    -   The resulting distribution is only a coarse approximation of the        optimal distribution, and, therefore, only a limited shaping        gain can be obtained.    -   The encoder for the m^(th) bit-level is more complex than a        conventional encoder.

In summary of the above, there is a need for an improved apparatus andmethod for encoding an input message into symbols.

SUMMARY

In view of the above-mentioned challenges and disadvantages, embodimentsof the present disclosure aim to improve the conventional approaches forencoding an input message into a symbol sequence. An objective isthereby to provide an apparatus and a method for multi-level encoding,which allow reducing the shaping loss. In particular, it should bepossible to provide the symbol sequence with a desired characteristic,e.g., with a predefined probability distribution of the symbols like aGaussian probability distribution.

The objective of the present disclosure is achieved by the solutionprovided in the enclosed independent claims. Advantageousimplementations of the present disclosure are further defined in thedependent claims.

According to a first aspect, an apparatus for multi-level encoding of aninput message into a symbol sequence comprising information bits isprovided. The apparatus is configured to divide the input message into aplurality of sub-messages; encode each of the sub-messages into acodeword, wherein a set of the sub-messages is encoded on the basis ofthe codewords obtained by encoding the sub-messages not in the set and apredefined function of the symbol sequence; and map the codewords intocorresponding symbols.

The set of the sub-messages includes one or more sub-messages. Thisprovides the advantage that the symbol sequence output by the apparatuscan be provided to have a predefined characteristic, e.g. a predefinedprobability distribution, like one that approximates a Gaussiandistribution. Thus, a large shaping gain is obtained. The shaping lossis therefore reduced, when transmitting the symbols sequence over achannel, e.g. an AWGN.

In an implementation form of the apparatus of the first aspect, thefunction is at least one of a probability distribution function of thesymbols; a norm associated to the symbol sequence.

In an implementation form of the apparatus of the first aspect, theapparatus is configured to encode the set of the sub-messages such thatafter a symbol mapping of the plurality of codewords to a plurality ofsymbols, a predetermined probability distribution of the symbols isobtained.

This can provide the advantage that the symbols have a probabilitydistribution, which approximates a Gaussian distribution, andaccordingly a large shaping gain is obtained.

In a further implementation form of the apparatus of the first aspect,the apparatus is configured to encode each sub-message in the set ofsub-messages by using a channel decoder.

In particular, the apparatus can encode each sub-message by using adevice, which contains a channel decoder, or which is realized by achannel decoder. That is, the channel decoder can be included in adevice. Using the channel decoder to encoder the sub-messages allowsencoding them based on the codewords obtained by encoding thesub-messages not in the set. Thereby, the encoding of the sub-messagesin the set can be influenced based on the predefined function of thesymbols sequence. This provides the advantage that the shaping loss canbe reduced.

In a further implementation form of the apparatus of the first aspect,the apparatus is further configured to allocate, to each sub-message inthe set of sub-messages, a sequence of shaping bits, wherein eachsequence of shaping bits is chosen based on the sub-message to which thesequence of shaping bits is allocated and on the codewords obtained byencoding the sub-messages not in the set.

In an implementation with polar codes, the sequence of shaping bits canbe transmitted in polar sub-channels, particularly in reliable polarsub-channels. In this way, they can have an influence on how theresulting codewords are generated. Using some of the reliablesub-channels for the transmission of the sequence of shaping bits mayreduce the transmission rate, but if chosen correctly they can have apositive effect on the resulting symbols (e.g. increase in the signal tonoise ratio) such that this positive effect causes a larger gain thanthe loss in the transmission rate.

Each sequence of shaping bits may further be chosen based on thepredefined function of the symbol sequence. The shaping bits provide theadvantage that a symbol sequence with desired characteristics can beobtained, e.g. with a predefined probability distribution. For instance,a good approximation of non-uniform probability distribution, e.g. aGaussian probability distribution can be achieved.

In a further implementation form of the apparatus of the first aspect,the apparatus is configured to map the codewords to symbols that havemultiple bit-levels.

In a further implementation form of the apparatus of the first aspect,the apparatus is configured to map the codewords in such a way that atleast one bit-level contains bits from only a particular codeword.

In a further implementation form of the apparatus of the first aspect,the at least one bit-level corresponds to a sign bit-level.

The sign bit-level is the bit-level, which defines the sign of theresulting symbol. If the resulting symbol is a complex number, thenthere are two bit-levels that define the sign of the real and theimaginary parts of the complex number.

In a further implementation form of the apparatus of the first aspect,the apparatus is configured to map the encoded sub-messages based onnatural binary labelling, Gray labelling, or set partitioning labelling.

In a further implementation form of the apparatus of the first aspect,the probability distribution of the symbols is a non-uniformdistribution.

This non-uniform distribution may be a Gaussian distribution. Thenon-uniform probability distribution provides the advantage that theshaping loss can be reduced.

In a further implementation form of the apparatus of the first aspect,the apparatus is configured to encode each sub-message in the set ofsub-messages on the basis of polar coding.

This provides the advantage that the polarization effects of polarcoding can be used in the encoding process.

In a further implementation form of the apparatus of the first aspect,the channel decoder is a polar decoder, for example a successivecancellation decoder, a list decoder, a belief propagation decoder or aflip decoder.

The channel decoder is in particular the one mentioned above. Thisprovides the advantage that different decoders with a low complexity canbe used.

In a further implementation form of the apparatus of the first aspect,the plurality of symbols are amplitude shift keying symbols orquadrature amplitude modulation symbols.

This provides the advantage that well known symbols can be used.

In a further implementation form of the apparatus of the first aspect,the apparatus is configured to provide the symbols to a receiver, and tofurther provide the receiver with at least one of the followingparameters using a separate channel than for providing the symbols: asize of a sequence of shaping bits included in the symbols, predefinedfunction of the symbol sequence, and a rule how the shaping bits areallocated.

According to a second aspect, the disclosure relates to a method formulti-level encoding of an input message into a symbol sequencecomprising information bits, the method comprising the steps of dividingthe input message into a plurality of sub-messages, encoding each of thesub-messages into a codeword, wherein a set of the sub-messages isencoded on the basis of the codewords obtained by encoding thesub-messages not in the set and a predefined function of the symbolsequence, and mapping the encoded sub-messages into correspondingsymbols.

Implementation forms of the method of the second aspect can be developedaccording to the implementation forms of the apparatus of the firstaspect. The method of the second aspect and its implementation formsprovide the same advantages as the apparatus of the first aspect and itsimplementation forms, respectively.

According to a third aspect, the disclosure relates to a computerprogram comprising a program code for performing the method of thesecond aspect when executed on a computer.

According to a fourth aspect, the disclosure relates to an apparatus formulti-level decoding, the apparatus configured to perform a demapping ofa sequence of input symbols based a predefined function of the symbolsequence to obtain a demapped sequence, wherein the input symbolsinclude encoded shaping bits, decode the demapped sequence, and discardthe decoded shaping bits.

The apparatus of the fourth aspect supports decoding according to theencoding scheme provided by the apparatus of the first aspect. Thus, theapparatus of the fourth aspect supports all the advantages mentionedabove. The apparatus of the first aspect may be a transmitter, and theapparatus of the fourth aspect may be a receiver. Together, theapparatus of the first aspect and the apparatus of the fourth aspect mayform a transmission system.

It has to be noted that all devices, elements, units and means describedin the present application could be implemented in the software orhardware elements or any kind of combination thereof. All steps whichare performed by the various entities described in the presentapplication as well as the functionalities described to be performed bythe various entities are intended to mean that the respective entity isadapted to or configured to perform the respective steps andfunctionalities. Even if, in the following description of specificembodiments, a specific functionality or step to be performed byexternal entities is not reflected in the description of a specificdetailed element of that entity which performs that specific step orfunctionality, it should be clear for a skilled person that thesemethods and functionalities can be implemented in respective software orhardware elements, or any kind of combination thereof.

BRIEF DESCRIPTION OF DRAWINGS

The above described aspects and implementation forms of the presentdisclosure will be explained in the following description of specificembodiments in relation to the enclosed drawings, in which:

FIG. 1 shows a schematic representation of a conventional encoder anddecoder.

FIG. 2 shows a schematic representation of a conventional encoder.

FIG. 3. shows a schematic representation of a conventional decoder.

FIG. 4 shows a schematic representation of a conventional encoder.

FIG. 5 shows a probability distribution of ASK symbols.

FIG. 6 shows a system comprising an encoder and a decoder communicatingvia a communication channel, according to embodiments of the disclosure.

FIG. 7 shows an apparatus for encoding an input message according to anembodiment of the disclosure.

FIG. 8 shows a probability distribution of ASK symbols obtained by anapparatus according to an embodiment of the disclosure.

FIG. 9 shows apparatus for encoding an input message according to anembodiment of the disclosure.

FIG. 10 shows a table containing information about bit mapping performedby an apparatus according to an embodiment of the disclosure.

FIG. 11 shows an apparatus for encoding an input message according to anembodiment of the disclosure.

FIG. 12 shows a method for encoding an input message according to anembodiment of the disclosure.

DETAILED DESCRIPTION OF EMBODIMENTS

Upon analyzing the existing solution it can be argued thatasymptotically (with infinitely long codewords), MLC performs betterthan BICM. The reason is that the single stage demapping used by BICMdoes not take into account the dependence between bit-levels, whereasthe multi-stage demapping of MLC allows full exploitation of thedependencies between bit-levels.

On the other hand, in the non-asymptotic regime, where the codewordshave finite lengths, MLC may show performance degradations. This is dueto finite length effects of the channel coding schemes: the performanceof channel codes degrades as the codeword lengths get smaller. MLCrequires multiple shorter length codewords (compared to a single longcodeword in BICM), and, hence, suffers from a larger finite length loss.Therefore, BICM is preferred in many communication systems.

In general, any binary channel code (e.g., turbo codes, LDPC codes,convolutional codes, polar codes) can be used for both BICM and MLC.Polar codes are the recently developed forward error correction schemes(i.e. channel coding schemes) that can achieve the capacity of binaryinput memoryless channels. However, their performance with BICM is oftenpoor compared to other modern coding schemes. On the other hand, it isknown that polar codes work well with MLC.

Polar coding relies on the channel polarization phenomenon, where thephysical channel is converted into polar sub-channels, which tend tohave either very high or very low reliabilities asymptotically. A polarencoder assigns message bits to reliable channels, and (known) frozenbits to unreliable channels. A polar decoder (such as a successivecancellation (SC) or SC List (SCL) decoder) processes a noisyobservation of the polar codeword together with the frozen bits, inorder to estimate the message bits.

Let G denote the polar transform matrix of size n by n, which is definedas the (log₂n)-th Kronecker power of the 2 by 2 kernel:

$F = {\begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}.}$

A polar codeword c is obtained from the input sequence u by c=uG. Here,u contains k message bits d at indices I, and n-k frozen bits at indicesF, where I and F denote the sets containing the indices of the polarsub-channels with high and low reliabilities, respectively. Theperformance of a polar code depends on the choice of the sets I and F.

In general, one can calculate the reliabilities of the polarsub-channels (for a given physical channel), and allocate the mostreliable sub-channels to message bits, and the rest for frozen bits. Asimpler approach is to use a polar sequence Q, similar to the polarsequence that is specified in the 5G New Radio specification. Such apolar sequence defines the reliability order of the polar sub-channels.For example, Q can include the sub-channel indices in order of ascendingreliabilities. Accordingly, I and F can be easily computed by taking thelast k and first n-k indices in Q, respectively.

The symbol mapper used in BICM or MLC schemes takes as input a sequenceof m-bits, and maps them to a channel input symbol according to abit-labeling scheme. For example, a mapper for 8-ASK maps 3 bits to oneASK symbol can take 8 possible values depending on the input bits.Usually, ‘Gray labeling’ is preferred for BICM, and ‘Natural BinaryLabeling’ or ‘Set Partitioning Labeling’ are preferred for MLC.

FIG. 6 shows a system 600 comprising an apparatus (or encoder) 601according to an embodiment, and an apparatus 603 (or decoder) accordingto an embodiment, which are configured to communicate via acommunication channel 602, e.g. via an AWGN.

The apparatus 601 is configured to perform multi-level encoding of aninput message into a symbol sequence, wherein the input message andsymbol sequence comprise information bits. The apparatus 601 isspecifically configured to perform the following steps:

-   -   Divide the input message into a plurality of sub-messages. Each        of the plurality of sub-messages may have the different lengths.    -   Encode each of the sub-messages into a codeword. Thereby, a set        of the sub-messages is encoded on the basis of the codewords        obtained by encoding the sub-messages not in the set and on the        basis of a predefined function of the symbols (i.e. of the        symbol sequence). That is, the sub-messages not in the set are        encoded, and the resulting codewords are provided as an input        for the encoding of the sub-messages in the set. The other input        for the encoding is the predefined function of the symbol        sequence, which may be a norm associated to the symbol sequence,        or a probability distribution function (or probability mass        function) of the symbols.    -   Map the encoded sub-messages into corresponding symbols. For        instance, the apparatus 601 may use a symbol mapper for        performing the mapping.

Thus, the symbol sequence may be provided with a predefinedcharacteristic, for instance, the symbols may have a predefinedprobability distribution (or a target probability distribution) afterthe symbol mapping. The symbol sequence may also fulfil a predefinednorm (for example Euclidean norm, p-norm, or a distance of the symbolsequence to a predetermined sequence).

After the apparatus 601 has encoded the input message, and has outputthe symbols, it can send the symbol sequence to the apparatus 603, whichis configured for multi-level decoding of an input symbol sequence. Thisapparatus 603 will be described in more detail further below.

FIG. 7 shows the apparatus 601 according to an embodiment of thedisclosure, which bases on the embodiment shown in FIG. 6. Inparticular, FIG. 7 shows the apparatus 601 with further, optionalfeatures. The apparatus 601 of FIG. 7 is configured to encode the inputmessage d into a symbol sequence of ASK symbols.

In this embodiment, the apparatus 601 is further configured to encodethe m^(th) bit-level (defining the set of sub-messages; related toencoder m) in dependence of the previous bit-levels (sub-messages not inthe set of sub-messages; related to encoders 1, 2). Further, theencoding in this embodiments may be in dependence of the predefinedfunction, e.g. a predefined probability distribution, which could bederived from the codewords of the previous bit-levels or and externalinput. As a result, for instance, a predefined probability distributionof the ASK symbols can be obtained, in particular, one that approximatesvery accurately a Gaussian distribution (see FIG. 8).

A decoder can be used as encoder to encode each sub-message in the setof sub-messages, where the decoder is configured to search for acodeword that represents the sub-message and at the same time possessescertain properties, such as a certain probability distribution (or aconditional probability distribution) of the bits in the codeword. Thiscan be realized for example as described in the work of Iscan et al“Probabilistically Shaped Multi-Level Coding with Polar Codes for FadingChannels,” In 2018 IEEE Globecom Workshops (GC Wkshps), pp. 1-5 withreference to polar codes.

Moreover, a simple successive cancellation (SC) decoder can be used asencoder to encode each sub-message in the set of sub-messages, i.e. hereas the encoder in the m^(th) bit-level, which has a much lowercomputational complexity compared to, for example, a list decoder. Infact, a SC decoder has the same order of complexity as a conventionalpolar encoder. Therefore, advantageously, the proposed scheme is notmore complex than a conventional MLC scheme based on polar codes.

In the following a summary of the steps is given, which may be performedby the apparatus 601 of FIG. 7:

-   -   The apparatus 601 is configured to use an MLC scheme, wherein        only a single bit-level may be (probabilistically) shaped.        However, the bit-level is encoded depending on the previously        encoded bit-levels. As a result, for example, the obtained        probability distribution of the ASK symbols can be determined,        e.g. to approximate very accurately a Gaussian distribution.        Thus, the shaping loss in the channel 602, particularly and        AWGN, can be significantly reduced.    -   The apparatus 601 can be configured to use an SC decoder for        encoding the m^(th) bit-level. This provides the advantage that        the apparatus 601 is also advantageous from a complexity        point-of-view.

The apparatus 601 is advantageous for reliable data transmission, isbased on multi-level coding (MLC), and may be further configured to:

-   -   Transmit polar codewords in each bit-level (or at least in one        bit-level, in particular at least in the m^(th) bit level). That        is, the apparatus 601 may be configured to encode each        sub-message in the set of sub-messages on the basis of polar        coding.    -   Generate in (at least) one bit-level (denoted as the shaped        bit-level), the codewords in dependence on the codewords from        other bit-levels, such that after the symbol mapping a        predefined probability distribution or target probability        distribution is obtained. That is, the apparatus 601 may be        further configured to allocate, to each sub-message in the set        of sub-messages, a sequence of shaping bits.    -   Use a polar decoder, in that bit-level, to perform the encoding.    -   Set the shaped bit-level as the sign bit-level, i.e., the        bit-level that defines the sign of the resulting symbols.

In the following, steps that can be performed by the apparatus 601 onthe transmitter side are further detailed.

The transmission scheme as given in FIG. 7 can be used, as well as asymbol mapper with e.g. a natural (binary) labeling. In an embodiment,for natural labeling with m bit-levels, the mapper output may takevalues {±1, ±3, ±5, . . . , ±(2^(m)−1)}. The mapper output may be scaledby a predefined constants, for example to satisfy power requirements.

For example, for 8-ASK (with m=3) the output symbols {−7, −5, . . . , 5,7} are possible. The mapping of each bit-sequence of m bits to the ASKsymbols may be performed according to the binary representation ofnatural numbers between 0 and (2^(m)−1) in ascending or descendingorder. The table in FIG. 10 gives an example for m=3, wherein three bitsb₁b₂b₃ are mapped to 8-ASK symbols x. Note that the last bit level (b₃)contains the sign-bit, i.e. depending on whether b₃ is one or zero, theresulting ASK symbol x has a negative or positive sign.

This scheme can easily be extended to symbols with complex numbers. Insuch a case, there will be two sign bit-levels in total (one signbit-level per complex dimension).

In an embodiment, such a symbol mapper can be implemented as given inthe dashed box of FIG. 11 for an example with m=3. Here, first thecodeword bits c_(i) (zeros and ones) are mapped to a sequence b_(i)containing ±1, which are then scaled by a_(i)=2^((i-1)). The sequence xthat contains the ASK symbols is obtained by summing up these scaledsequences. The cumulative sum of the scaled sequences of the bit levels1 to i can be noted as x_(i).

As described above, the encoder in the m^(th) bit-level (i.e. the signbit-level) is a modified encoder. This encoder is denoted as ε_(s). Inan embodiment, a polar decoder is used as the encoder, e.g. a successivecancellation (SC) decoder, a successive cancellation list (SCL) decoder,a belief propagation (BP) decoder, or a flip decoder can be employed.Moreover, the encoder ε_(s) generates its output dependent on x_(m-1).Note that x_(m-1) is constructed based on the codewords in thebit-levels 1 to m−1. Therefore, the output of ε_(s) is dependent on thecodewords in all previous bit-levels.

More specifically, the encoder ε_(s) looks for a codeword c_(m) thatrepresents the m^(th) message part d_(m) (i.e. ε_(s) encodes d_(m)), andat the same time causes the ASK symbols in x to be distributed accordingto the target probability distribution P_(X). For this purpose, some ofthe reliable polar sub-channels can be allocated in the m^(th) bit-levelfor shaping bits, which do not carry any information, but cause x to bedistributed according to the target probability distribution P_(X).

The number of shaping bits (s) defines how much resources (in this casepolar sub-channels) are allocated for signal shaping. In conventionalschemes without shaping no resources are allocated for shaping, i.e.s=0. The number of shaping bits can be chosen to get the bestapproximation of the target distribution. On the other hand, eachshaping bit uses an additional resource. Therefore, too many shapingbits should not be used. The optimal number of shaping bits is theminimum number that gives the maximum gain.

If k_(m) message bits are to be transmitted on the m^(th) bit-level(i.e. the length of d_(m) is k_(m)), and s shaping bits are to be used,assuming a fixed polar sequence, the most reliable s polar sub-channels(described by the set S) are allocated for the shaping bits, the nextmost reliable k_(m) polar sub-channels (described by the set I) formessage bits, and the rest (denoted by the set F) for frozen bits.

Under these conditions, a polar decoder as ε_(s) (with codeword length nand rate (k_(m)+s)/n) can be used, wherein the following parameters areemployed:

-   -   The known bits (e.g. zeros) can be used as frozen bits at        indices F.    -   The message bits d_(m) can be used as additional frozen bits at        indices I.    -   S can be used as the indices of the unknown bits (to be        recovered by the decoder).    -   Λ (defined below) can be used as the decoder input in        log-likelihood ratio (LLR) form (noisy channel observation).

In general, Λ can be defined as a function of x_(m-1). Hence, it is alsoa function of the codeword bits from the previous bit-levels. Λ isproportional to −x_(m-1) for a Maxwell-Boltzmann target distributionthat minimizes the transmit power for a given rate. Therefore,Λ=−x_(m-1) can be used. Moreover, a simple successive cancellation (SC)decoder can be sufficient to obtain good gains. A more complex SCLdecoder results in a better performance, but at a cost of increasedcomplexity.

Note that this polar decoding operation would search for the shapingbits (and hence the resulting codeword c_(m)), which would cause theresulting codeword to have the desired probability distributionconditioned on x_(m-1). FIG. 8 shows the resulting distribution of theASK symbols, where m=4 and roughly ⅓ of the polar sub-channels in them^(th) bit-level is allocated for shaping bits.

From a different perspective, this polar decoding operation can also beseen as the solution of an energy minimization problem, i.e. the decoderlooks for a codeword, such that the resulting ASK symbols have a minimumaverage energy, i.e. minimum Euclidean norm.

To sum up, the following features can be considered for the apparatus601:

-   -   An MLC scheme can be used based on polar codes, where polar        codewords are used in each bit-level.    -   A natural (binary) labelling can be used.    -   A modified encoder ε_(s) can be used in the m^(th) bit-level        (which corresponds to the sign bit-level).    -   The codeword c_(m) can be generated by the encoder ε_(s) in        dependence on the codewords from previous bit-levels (c₁ to        c_(m-1)).    -   The encoder ε_(s) can be implemented by using a polar decoder,        e.g., by a simple SC decoder or more complex SCL decoder.    -   The codeword in the m^(th) bit-level can contain (besides        message and frozen bits) shaping bits, which do not carry any        additional information, but cause the resulting codeword to have        a probability distribution conditioned on the bits of the        previously encoded codewords, which further results in a target        or first probability distribution P_(X) of the ASK symbols after        symbol mapping.    -   The number of shaping bits s is a parameter, which can be chosen        to optimize the performance: if no shaping bits are used (s=0),        then no shaping gain can be obtained. If too many shaping bits        are used, then it can lead to an inefficient usage of the        available resources. Our findings indicate that using roughly ⅓        of the polar sub-channels in the sign-bit level is a good        choice.

FIG. 9 shows the apparatus 601 according to an embodiment of thedisclosure, which bases on the embodiment shown in FIG. 7. Inparticular, FIG. 9 shows the apparatus 601 with further, optionalfeatures. The apparatus 601 of FIG. 9 is configured to encode the inputmessage into a symbol sequence, e.g. of ASK symbols.

In this embodiment, the apparatus 601 is configured to encode the m^(th)bit-level (defining the set of sub-messages; related to encoder m) independence of the previous bit-levels (sub-messages not in the set ofsub-messages; related to encoders 1, 2 and in dependence of thepredefined function (external input), e.g. a predefined probabilitydistribution. As a result, for instance, a predefined probabilitydistribution of the ASK symbols can be obtained, in particular, one thatapproximates very accurately a Gaussian distribution (see FIG. 8). Inparticular, a channel decoder may be used to encode the m^(th)bit-level.

In general, an encoder maps a message sequence to a codeword sequence.The operation is one-to-one, i.e., for each message sequence, there isanother codeword sequence. In general, codeword sequences are longerthan the message sequences. For example, assume an encoder that maps abinary message sequence of k bits to a codeword sequence of n bits withn>k. In this case, any binary sequence of length k can be the input ofthe encoder (there are 2^(k) possibilities of length k binarysequences), and there are 2^(k) possible codeword sequences of length n.Note that in general, there are 2^(n) different binary sequences oflength n, but not every length-n sequence is a codeword. The set of allpossible codeword sequence is called a codebook.

In this context, a channel decoder is a device that takes as input anunconstrained sequence of length n, and looks for a codeword sequence inthe codebook, and its corresponding message sequence. Both encoders anddecoders define mappings between sequences. Conventionally, encoders areused to generate codewords at the transmitters, and decoders are used tosearch for most likely codeword (and its corresponding message sequence)given the received noisy signal at the receiver.

In the apparatus 601 according to an embodiment of the disclosure, a(channel) decoder may be used, such that the codeword found by thischannel decoder has two properties: it represents the message, and atthe same time it has certain desired properties (i.e., the bits in thecodeword are distributed according to a desired probabilitydistribution, or desired conditional probability distribution). This maybe realized by introducing shaping bits as described above, which give anew degree of freedom for mapping the message sequence to a codeword.Shaping bits can be seen as additional bits (to be appended to messagebits) that do not carry any information, but cause the codeword bits tohave the desired probability distribution. How to obtain the values ofthe shaping bits can be formulated as a channel decoding operation (asdescribed in the work of Iscan et al). Therefore in the problem at hand,a channel decoder can be used in the apparatus 601 instead of anencoder.

After the apparatus 601 (of any one of the above described) has encodedthe input message, and has output the symbol sequence, it can send thesymbol sequence over the channel 602 to the apparatus 603 (see FIG. 1),which is configured for multi-level decoding of an input symbolsequence. This apparatus 603 may, for example, be configured, to:

-   -   Perform a demapping of the sequence of input symbols based on a        predefined function of the symbol sequence (e.g., based on the        predefined function of the apparatus 601, obtained from the        apparatus 601 or from another entity) to obtain a demapped        sequence, wherein the input symbols include encoded shaping        bits.    -   Decode the demapped sequence.    -   Discard the decoded shaping bits.

In particular, the apparatus 603 may be based on an MLC receiver asdepicted in FIG. 3. However, certain modifications may be performed.Particularly, compared to a conventional MLC receiver, the followingparameters are preferably known at the apparatus 603:

-   -   The predefined function of the symbol sequence, e.g., the        probability distribution function P_(X) of the ASK symbols.    -   The number of shaping bits s.    -   The set S, which indicates the indices of the shaping bits.

In general, all these parameters may be signaled to the apparatus 603(e.g., using a control channel), such that the apparatus 603 can usethese parameters to revert the operations performed at the apparatus601. However, some simplifications can be made, because all theseparameters are related to each other.

In the embodiment above, a fixed polar sequence has been assumed, andthe most reliable s indices have been used in this sequence in order tobuild S. Therefore, in such a situation the apparatus 603 can alreadyobtain S, if the number of shaping bits s is known. Moreover, usingroughly ⅓ of the polar sub-channels for shaping bits is a good choice.Accordingly, a fixed rule can be used in order to obtain the number ofshaping bits. Lastly, since s and P_(X) are related, the resulting P_(X)for each choice of s can be precomputed, and stored in a look-up table,such that additional control signaling is avoided.

Conventional MLC receivers assume that the ASK symbols are uniformlydistributed (P_(X) is uniform). In an embodiment, P_(X) is non-uniform,and, therefore, the demapper needs to generate its output depending onP_(X). For some typical distributions (like Gaussian distribution orMaxwell-Boltzmann distribution), this can be accomplished by scaling thedemapper inputs by a constant, where the constant depends on P_(X) andthe channel noise variance.

Compared to a conventional MLC receiver, there are additional shapingbits that are allocated in some polar sub-channels in the signbit-level. The values of the shaping bits are unknown to the apparatus603. During decoding of this bit-level, the decoder can treat theshaping bits as the message bits, which are also unknown. At the end ofthe decoding process, the apparatus 603 can basically discard theshaping bits as they do not carry any additional information. Moreover,if all of the message bits are recovered during decoding, the decodercan perform an early termination without finishing the whole decodingprocess (since the remaining unknown bits are only shaping bits). Asanother alternative, the apparatus 603 can extract all shaping bits andmessage bits at the output of the decoder, and calculate another copy ofthe shaping bits based on the message bits (as it is done at thetransmitter). Later, the apparatus 603 can compare this copy of theshaping bits and the shaping bits at the output of the decoder, and ifthey are not identical, the apparatus 603 can declare an error. This canbe seen as an additional error detection mechanism, which can also beused for picking the correct codeword from the output of a list decoder.

To sum up, in embodiments of the disclosure, the following features maybe implemented at the receiver side apparatus 603:

-   -   The probability distribution P_(X) of the ASK symbols, the        number of shaping bits s and the set S are the additional        parameters, which are used by the apparatus 603 to recover the        transmitted message. Therefore, they may be signalled to the        apparatus 603 (e.g., using a control or communication channel        other than the channel 602). However, as they are related to        each other, one can only signal a subset of them, such that the        other parameters are obtained depending on these parameters.        Alternatively, one uses fixed rules to obtain these parameters,        such that no additional signalling is required.    -   The demapper may use P_(X) during demapping. For some typical        probability distributions, this can be accomplished by scaling        the demapper inputs by a scalar factor that depends on P_(X) and        the channel noise variance.    -   The apparatus 603 may treat the shaping bits as if they were        message bits. After the decoding is finished, the apparatus 603        can discard the shaping bits, or use them as an error detection        mechanism. Alternatively, the decoder can also perform        early-termination if all message bits are decoded.

FIG. 12 shows a method 1200 for encoding an input message into a symbolsequence, according to an embodiment. The method 1200 may be performedby the apparatus 601. The method 1200 for multi-level encoding of theinput message, which comprises information bits, comprises the steps of:

-   -   Dividing 1201 the input message into a plurality of        sub-messages.    -   Encoding 1202 each of the sub-messages into a codeword, wherein        a set of the sub-messages is encoded on the basis of the        codewords obtained by encoding the sub-messages not in the set        and a predefined function of the symbol sequence.    -   Mapping 1203 the encoded sub-messages into corresponding        symbols.

The present disclosure has been described in conjunction with variousembodiments as examples as well as implementations. However, othervariations can be understood and effected by those persons skilled inthe art and practicing the claimed disclosure, from the studies of thedrawings, this disclosure and the independent claims. In the claims aswell as in the description the word “comprising” does not exclude otherelements or steps and the indefinite article “a” or “an” does notexclude a plurality. A single element or other unit may fulfill thefunctions of several entities or items recited in the claims. The merefact that certain measures are recited in the mutual different dependentclaims does not indicate that a combination of these measures cannot beused in an advantageous implementation.

What is claimed is:
 1. An apparatus for multi-level encoding of an inputmessage into a symbol sequence comprising information bits, theapparatus comprising at least one processor and one or more memoriescoupled to the at least one processor, the one or more memoriescomprising program codes that, when executed by the at least oneprocessor, cause the apparatus to: divide the input message into aplurality of sub-messages; encode each of the sub-messages into acodeword, wherein a set of the sub-messages is encoded on the basis ofthe codewords obtained by encoding the sub-messages not in the set and apredefined function of the symbol sequence; and map the codewords intocorresponding symbols.
 2. The apparatus according to claim 1, whereinthe function is at least one of a probability distribution function ofthe symbols and a norm associated to the symbol sequence.
 3. Theapparatus according to claim 1, wherein the program codes, when executedby the at least one processor, cause the apparatus to encode eachsub-message in the set of sub-messages by using a channel decoder. 4.The apparatus according to claim 1, wherein the program codes, whenexecuted by the at least one processor, cause the apparatus to allocate,to each sub-message in the set of sub-messages, a sequence of shapingbits, wherein each sequence of shaping bits is chosen based on thesub-message to which the sequence of shaping bits is allocated and onthe codewords obtained by encoding the sub-messages not in the set. 5.The apparatus according to claim 1, wherein the program codes, whenexecuted by the at least one processor, cause the apparatus to map thecodewords to symbols that have multiple bit-levels.
 6. The apparatusaccording to claim 5, wherein the program codes, when executed by the atleast one processor, cause the apparatus to map the codewords in such away that at least one bit-level contains bits from only a particularcodeword.
 7. The apparatus according to claim 6, wherein the at leastone bit-level corresponds to a sign bit-level.
 8. The apparatusaccording to claim 1, wherein the program codes, when executed by the atleast one processor, cause the apparatus to map the encoded sub-messagesbased on natural binary labelling, Gray labelling, or set partitioninglabelling.
 9. The apparatus according to claim 1, wherein a probabilitydistribution of the symbols is a non-uniform distribution.
 10. Theapparatus according to claim 1, wherein the program codes, when executedby the at least one processor, cause the apparatus to encode eachsub-message in the set of sub-messages on the basis of polar coding. 11.The apparatus according to claim 3, wherein the channel decoder is apolar decoder.
 12. The apparatus according to claim 1, wherein thesymbols are amplitude shift keying symbols or quadrature amplitudemodulation symbols.
 13. The apparatus according to claim 1, wherein theprogram codes, when executed by the at least one processor, cause theapparatus to provide the symbols to a receiver, and to further providethe receiver with at least one of the following parameters using aseparate channel than for providing the symbols: a size of a sequence ofshaping bits included in the symbols; the predefined function of thesymbol sequence; and an indicator or a rule how the shaping bits areallocated.
 14. Method or multi-level encoding of an input message into asymbol sequence comprising information bits, the method comprising thesteps of: dividing the input message into a plurality of sub-messages;encoding each of the sub-messages into a codeword, wherein a set of thesub-messages is encoded on the basis of the codewords obtained byencoding the sub-messages not in the set and a predefined function ofthe symbol sequence; and mapping the encoded sub-messages intocorresponding symbols.
 15. The method according to claim 14, furthercomprises: allocating, to each sub-message in the set of sub-messages, asequence of shaping bits, wherein each sequence of shaping bits ischosen based on the sub-message to which the sequence of shaping bits isallocated and on the codewords obtained by encoding the sub-messages notin the set.
 16. The method according to claim 14, further comprises:mapping the codewords to symbols that have multiple bit-levels.
 17. Themethod according to claim 16, further comprises: mapping the codewordsin such a way that at least one bit-level contains bits from only aparticular codeword.
 18. The method according to claim 14, furthercomprises: mapping the encoded sub-messages based on natural binarylabelling, Gray labelling, or set partitioning labelling.
 19. The methodaccording to claim 14, wherein the program codes, when executed by theat least one processor, cause the apparatus to provide the symbols to areceiver, and to further provide the receiver with at least one of thefollowing parameters using a separate channel than for providing thesymbols: a size of a sequence of shaping bits included in the symbols;the predefined function of the symbol sequence; and an indicator or arule how the shaping bits are allocated.
 20. An apparatus formulti-level decoding, comprising at least one processor and one or morememories coupled to the at least one processor, the one or more memoriescomprising program codes that when executed by the at least oneprocessor, cause the apparatus to: perform a demapping of a sequence ofinput symbols based on a predefined function of the symbol sequence toobtain a demapped sequence, wherein the input symbols include encodedshaping bits; decode the demapped sequence; and discard the decodedshaping bits.